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HMMT 十一月 2009 · 冲刺赛 · 第 15 题

HMMT November 2009 — Guts Round — Problem 15

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

  1. [ 8 ] The curves x + y = 36 and y = x − 7 intersect at four points. Find the sum of the squares ofthe x -coordinates of these points.
    2 ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 7 NOVEMBER 2009 — GUTS ROUNDnd 1

英文原题

[ 8 ] The curves x 2 + y 2 = 36 and y = x 2 − 7 intersect at four points. Find the sum of the squares of
the x -coordinates of these points.
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2 nd ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 7 NOVEMBER 2009 — GUTS ROUND

解析

英文解析

  1. [ 8 ] The curves x + y = 36 and y = x − 7 intersect at four points. Find the sum of the squares ofthe x -coordinates of these points.
    2 2
    Answer: 26 If we use the system of equations to solve for y , we get y + y − 29 = 0 (since x = y +7).
    The sum of the roots of this equation is − 1. Combine this with x = y + 7 to see that the sum of the 2
    square of the possible values of x is 2 · ( − 1 + 7 · 2) = 26.
    2 ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 7 NOVEMBER 2009 — GUTS ROUNDnd 1